概率赋范线性空间中全连续线性算子的谱定理

Spectral theory of completely continuous linear operators in probabilistic normcd spaces

本文证明Menger概率赋范线性空间X中的F.Riesz引理。若T是X中的全连续漤线性算子,θ≠0,T_θ=T-θI的零空间为N(T_θ),则dimN(T_θ)<∞.T的固有值至多可数,并以0为仅有的可能聚点.T_θ的值域R(T_θ)是闭的.T的非零谱值都是T的固有值。

In this paper we prove F-Riesz's Lemma in Menger probabilistic normedspace X. Let T be a completely continuous linear operator on X, θ≠θ, the nullspace of Tθ=T-θI is N(Tθ), then dim N(Tθ)<∞. The set of eigenvalues of Tis at most countable , and the possible point of accumulation is only θ = θ. Therange R(Tθ} of Tθ is closed. Every spectral value θ≠θ of T is an eigenvalueof T.